The continuous real functions $f:\mathbb R\to\mathbb R$, which fulfill the functional equation $f(x+y)=f(x)+f(y),$ are exactly the linear functions $f(x)=ax$ for some $a\in\mathbb R.$
The continuous real functions $f:\mathbb R_+^*\to\mathbb R$, which fulfill the functional equation $f(xy)=f(x)+f(y),$ are exactly the logarithmic functions $f(x)=a\ln(x)$ for some $a\in\mathbb R.$
The continuous real functions $f:\mathbb R_+^*\to\mathbb R$, which fulfill the functional equation $f(xy)=f(x)f(y),$ are either the constant zero function or the exponential functions $f(x)=x^a$ for some $a\in\mathbb R.$
Proofs: 1